On the restricted $k$-multipartition function

Let $\mathbf{a}=(a_1,\dots ,a_r)$ be a sequence of positive integers and $k\ge 2$ an integer. We study $p_{k,\mathbf{a}}(n)$, the restricted $k$-multipartition function associated to $\mathbf{a}$ and $k$. We prove new formulas for $p_{k,\mathbf{a}}(n)$, its waves $W_j(n,k,\mathbf{a})$'s and its polynomial part $P_{k,\mathbf{a}}(n)$. Also, we give a lower bound for the density of the set $\{n\ge 0:p_{k,\mathbf{a}}(n)\equiv ̸0(m)\}$, where $m\ge 2$ is an integer.